The paper deals with the result presented also in the author's paper [Set-Valued Anal. 9, No. 1--2, 159--168 (2001; Zbl 1006.52001)]: under specified conditions an image created by a nonlinear map of a small ball in the Hilbert space is a convex set. This result has numerous applications in linear algebra, mathematical programming and control systems. The author proves convexity for a set of all eigenvalues of a family of perturbed matrices, for a set of all zeros of a certain family of polynomials and for a value set of a family of determinants (so-called \(\mu\)-analysis). In the next section the author proves necessary and sufficient conditions for optimality of a certain local mathematical programming problem and convergence of a special iterative method. In the last section of the paper a brief survey -- without proofs -- of applications in control theory is presented: a convexity of the reachable set of a nonlinear control system under some assumptions, optimality conditions for some convex optimal control problem and an iterative solution method. At last analogous results are mentioned for the discrete time case. The paper is written in a clear and precise manner. It may be expected that much more applications can arise in other fields, including functional and numerical analysis.